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Social ContagionPrinciples of Complex Systems

CSYS/MATH 300, Spring, 2013 | #SpringPoCS2013

Prof. Peter Dodds@peterdodds

Department of Mathematics & Statistics | Center for Complex Systems |Vermont Advanced Computing Center | University of Vermont

Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License.

Social Contagion

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These slides brought to you by:

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Outline

Social Contagion ModelsBackgroundGranovetter’s modelNetwork versionFinal sizeSpreading successGroups

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Social Contagion

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Things that spread well:

buzzfeed.com ():

+ News ...

Social Contagion

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LOL + cute + fail + wtf:

Social Contagion

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The whole lolcats thing:

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Some things really stick:

Social Contagion

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wtf + geeky + omg:

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Social Contagion

http://xkcd.com/610/ ()

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Social Contagion

Social Contagion

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Social Contagion

Social Contagion

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Social Contagion

Examples abound

I fashionI strikingI smoking () [7]

I residentialsegregation [19]

I ipodsI obesity () [6]

I Harry PotterI votingI gossip

I Rubik’s cubeI religious beliefsI leaving lectures

SIR and SIRS contagion possibleI Classes of behavior versus specific behavior: dieting

Social Contagion

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Framingham heart study:

Evolving network stories (Christakis and Fowler):I The spread of quitting smoking () [7]

I The spread of spreading () [6]

I Also: happiness () [9], loneliness, ...I The book: Connected: The Surprising Power of Our

Social Networks and How They Shape Our Lives ()

Controversy:I Are your friends making you fat? () (Clive

Thomspon, NY Times, September 10, 2009).I Everything is contagious ()—Doubts about the

social plague stir in the human superorganism (DaveJohns, Slate, April 8, 2010).

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Social Contagion

Two focuses for usI Widespread media influenceI Word-of-mouth influence

We need to understand influenceI Who influences whom? Very hard to measure...I What kinds of influence response functions are

there?I Are some individuals super influencers?

Highly popularized by Gladwell [10] as ‘connectors’I The infectious idea of opinion leaders (Katz and

Lazarsfeld) [16]

Social Contagion

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The hypodermic model of influence

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The two step model of influence [16]

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The general model of influence

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Social Contagion

Why do things spread?I Because of properties of special individuals?I Or system level properties?I Is the match that lights the fire important?I Yes. But only because we are narrative-making

machines...I We like to think things happened for reasons...I Reasons for success are usually ascribed to intrinsic

properties (e.g., Mona Lisa)I System/group properties harder to understandI Always good to examine what is said before and

after the fact...

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The Mona Lisa

I “Becoming Mona Lisa: The Making of a GlobalIcon”—David Sassoon

I Not the world’s greatest painting from the start...I Escalation through theft, vandalism, parody, ...

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The completely unpredicted fall of EasternEurope

Timur Kuran: [17, 18] “Now Out of Never: The Element ofSurprise in the East European Revolution of 1989”

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The dismal predictive powers of editors...

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Social Contagion

Messing with social connectionsI Ads based on message content

(e.g., Google and email)I BzzAgent ()I One of Facebook’s early advertising attempts:

Beacon ()I All of Facebook’s advertising attempts.

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Getting others to do things for youA very good book: ‘Influence’ [8] by Robert Cialdini ()

Six modes of influence:1. Reciprocation: The Old Give and Take... and Take;

e.g., Free samples, Hare Krishnas.2. Commitment and Consistency: Hobgoblins of the

Mind ; e.g., Hazing.3. Social Proof: Truths Are Us;

e.g., Jonestown (),Kitty Genovese () (contested).

4. Liking: The Friendly Thief ; e.g., Separation intogroups is enough to cause problems.

5. Authority: Directed Deference;e.g., Milgram’s obedience to authorityexperiment. ()

6. Scarcity: The Rule of the Few ; e.g., Prohibition.

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Social contagion

I Cialdini’s modes are heuristics that help up us getthrough life.

I Useful but can be leveraged...

Other acts of influence:I Conspicuous Consumption (Veblen, 1912)I Conspicuous Destruction (Potlatch)

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Social Contagion

Some important models:I Tipping models—Schelling (1971) [19, 20, 21]

I Simulation on checker boardsI Idea of thresholdsI Explore the Netlogo () online

implementation () [26]

I Threshold models—Granovetter (1978) [13]

I Herding models—Bikhchandani, Hirschleifer, Welch(1992) [2, 3]

I Social learning theory, Informational cascades,...

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Social contagion models

ThresholdsI Basic idea: individuals adopt a behavior when a

certain fraction of others have adoptedI ‘Others’ may be everyone in a population, an

individual’s close friends, any reference group.I Response can be probabilistic or deterministic.I Individual thresholds can varyI Assumption: order of others’ adoption does not

matter... (unrealistic).I Assumption: level of influence per person is uniform

(unrealistic).

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Social Contagion

Some possible origins of thresholds:I Inherent, evolution-devised inclination to coordinate,

to conform, to imitate. [1]

I Lack of information: impute the worth of a good orbehavior based on degree of adoption (social proof)

I Economics: Network effects or network externalitiesI Externalities = Effects on others not directly involved

in a transactionI Examples: telephones, fax machine, Facebook,

operating systemsI An individual’s utility increases with the adoption level

among peers and the population in general

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Threshold models—response functions

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

φ

p

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

φ

p

I Example threshold influence response functions:deterministic and stochastic

I φ = fraction of contacts ‘on’ (e.g., rioting)I Two states: S and I.

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Social Contagion

Granovetter’s Threshold model—definitionsI φ∗ = threshold of an individual.I f (φ∗) = distribution of thresholds in a population.I F (φ∗) = cumulative distribution =

∫ φ∗φ′∗=0 f (φ′∗)dφ′∗

I φt = fraction of people ‘rioting’ at time step t .

I At time t + 1, fraction rioting = fraction with φ∗ ≤ φt .I

φt+1 =

∫ φt

0f (φ∗)dφ∗ = F (φ∗)|φt

0 = F (φt )

I ⇒ Iterative maps of the unit interval [0,1].

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Threshold models

Action based on perceived behavior of others:

0 10

0.2

0.4

0.6

0.8

1

φi∗

A

φi,t

Pr(a

i,t+

1=1)

0 0.5 10

0.5

1

1.5

2

2.5B

φ∗

f (φ∗ )

0 0.5 10

0.2

0.4

0.6

0.8

1

φt

φ t+1 =

F (

φ t)

C

I Two states: S and I.I φ = fraction of contacts ‘on’ (e.g., rioting)I Discrete time update (strong assumption!)I This is a Critical mass model

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Threshold models

Another example of critical mass model:

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

γ

f(γ)

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

φt

φ t+1

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Threshold models

Example of single stable state model:

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

γ

f(γ)

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

φt

φ t+1

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Threshold models

Chaotic behavior possible [15, 14]

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

xn

F (

x n+1 )

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

xn

F (

x n+1 )

I Period doubling arises as map amplitude r isincreased.

I Synchronous update assumption is crucial

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Threshold models—Nutshell

Implications for collective action theory:1. Collective uniformity 6⇒ individual uniformity2. Small individual changes⇒ large global changes

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“A simple model of global cascades on randomnetworks”I Many years after Granovetter and Soong’s work: D.

J. Watts. Proc. Natl. Acad. Sci., 2002 [23]

I Mean field model→ network modelI Individuals now have a limited view of the world

We’ll also explore:I “Seed size strongly affects cascades on random

networks” [12]

Gleeson and Cahalane, Phys. Rev. E, 2007.I “Influentials, Networks, and Public Opinion

Formation” [24]

Watts and Dodds, J. Cons. Res., 2007.I “Threshold models of Social Influence” [25]

Watts and Dodds, The Oxford Handbook ofAnalytical Sociology, 2009.

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Threshold model on a network

I Interactions between individuals now represented bya network

I Network is sparseI Individual i has ki contactsI Influence on each link is reciprocal and of unit weightI Each individual i has a fixed threshold φi

I Individuals repeatedly poll contacts on networkI Synchronous, discrete time updatingI Individual i becomes active when

fraction of active contacts aiki≥ φi

I Individuals remain active when switched (norecovery = SI model)

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Threshold model on a network

t=1 t=2 t=3

c

a

bc

e

a

b

e

a

bc

e

d dd

I All nodes have threshold φ = 0.2.

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Snowballing

First study random networks:I Start with N nodes with a degree distribution pk

I Nodes are randomly connected (carefully so)I Aim: Figure out when activation will propagateI Determine a cascade condition

The Cascade Condition:1. If one individual is initially activated, what is the

probability that an activation will spread over anetwork?

2. What features of a network determine whether acascade will occur or not?

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Example random network structure:

I Ωcrit = Ωvuln =critical mass =globalvulnerablecomponent

I Ωtrig =triggeringcomponent

I Ωfinal =potential extentof spread

I Ω = entirenetwork

Ωcrit ⊂ Ωtrig; Ωcrit ⊂ Ωfinal; and Ωtrig,Ωfinal ⊂ Ω.

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Snowballing

Follow active linksI An active link is a link connected to an activated

node.I If an infected link leads to at least 1 more infected

link, then activation spreads.I We need to understand which nodes can be

activated when only one of their neigbors becomesactive.

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The most gullible

Vulnerables:I We call individuals who can be activated by just one

contact being active vulnerablesI The vulnerability condition for node i :

1/ki ≥ φi

I Which means # contacts ki ≤ b1/φicI For global cascades on random networks, must have

a global cluster of vulnerables [23]

I Cluster of vulnerables = critical massI Network story: 1 node→ critical mass→ everyone.

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Cascade condition

Back to following a link:I A randomly chosen link, traversed in a random

direction, leads to a degree k node with probability∝ kPk .

I Follows from there being k ways to connect to anode with degree k .

I Normalization:∞∑

k=0

kPk = 〈k〉

I SoP(linked node has degree k) =

kPk

〈k〉

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Cascade condition

Next: Vulnerability of linked nodeI Linked node is vulnerable with probability

βk =

∫ 1/k

φ′∗=0f (φ′∗)dφ′∗

I If linked node is vulnerable, it produces k − 1 newoutgoing active links

I If linked node is not vulnerable, it produces no activelinks.

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Cascade condition

Putting things together:I Expected number of active edges produced by an

active edge:

R =∞∑

k=1

(k − 1) · βk ·kPk

〈k〉︸ ︷︷ ︸success

+ 0 · (1− βk ) · kPk

〈k〉︸ ︷︷ ︸failure

=∞∑

k=1

(k − 1) · βk ·kPk

〈k〉

Social Contagion

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Cascade condition

So... for random networks with fixed degree distributions,cacades take off when:

∞∑k=1

(k − 1) · βk ·kPk

〈k〉≥ 1.

I βk = probability a degree k node is vulnerable.I Pk = probability a node has degree k .

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Cascade condition

Two special cases:I (1) Simple disease-like spreading succeeds: βk = β

β ·∞∑

k=1

(k − 1) · kPk

〈k〉≥ 1.

I (2) Giant component exists: β = 1

1 ·∞∑

k=1

(k − 1) · kPk

〈k〉≥ 1.

Social Contagion

Social ContagionModelsBackground

Granovetter’s model

Network version

Final size

Spreading success

Groups

References

i

ϕ = 1/3

t=4 = active at t=0

= active at t=1

= active at t=2

= active at t=3

= active at t=4

50 of 94

Cascades on random networks

1 2 3 4 5 6 70

0.2

0.4

0.6

0.8

1

z

⟨ S

⟩

Example networks

PossibleNo

Cascades

Low influence

Fraction ofVulnerables

cascade sizeFinal

CascadesNo Cascades

CascadesNo

High influence

I Cascades occuronly if size of maxvulnerable cluster> 0.

I System may be‘robust-yet-fragile’.

I ‘Ignorance’facilitatesspreading.

Social Contagion

Social ContagionModelsBackground

Granovetter’s model

Network version

Final size

Spreading success

Groups

References

i

ϕ = 1/3

t=4 = active at t=0

= active at t=1

= active at t=2

= active at t=3

= active at t=4

51 of 94

Cascade window for random networks

1 2 3 4 5 6 70

0.2

0.4

0.6

0.8

1

z

⟨ S

⟩

0.05 0.1 0.15 0.2 0.250

5

10

15

20

25

30

φ

z

cascades

no cascades

infl

uenc

e

= uniform individual threshold

I ‘Cascade window’ widens as threshold φ decreases.I Lower thresholds enable spreading.

Social Contagion

Social ContagionModelsBackground

Granovetter’s model

Network version

Final size

Spreading success

Groups

References

i

ϕ = 1/3

t=4 = active at t=0

= active at t=1

= active at t=2

= active at t=3

= active at t=4

52 of 94

Cascade window for random networks

Social Contagion

Social ContagionModelsBackground

Granovetter’s model

Network version

Final size

Spreading success

Groups

References

i

ϕ = 1/3

t=4 = active at t=0

= active at t=1

= active at t=2

= active at t=3

= active at t=4

53 of 94

All-to-all versus random networks

0

0.2

0.4

0.6

0.8

1

a0 a

t

F (

a t+1)

all−to−all networks

A

0

0.2

0.4

0.6

0.8

1

⟨ k ⟩

⟨ S ⟩

random networks

B

0 0.5 10

0.2

0.4

0.6

0.8

1

a0 a’

0a

t

F (

a t+1)

C

0 10 200

0.2

0.4

0.6

0.8

1

⟨ k ⟩

⟨ S ⟩

D

0 0.5 10

5

10

φ∗

f (φ ∗)

0 0.5 10

2

4

φ∗

f (φ ∗)

Social Contagion

Social ContagionModelsBackground

Granovetter’s model

Network version

Final size

Spreading success

Groups

References

i

ϕ = 1/3

t=4 = active at t=0

= active at t=1

= active at t=2

= active at t=3

= active at t=4

54 of 94

Cascade window—summary

For our simple model of a uniform threshold:1. Low 〈k〉: No cascades in poorly connected networks.

No global clusters of any kind.2. High 〈k〉: Giant component exists but not enough

vulnerables.3. Intermediate 〈k〉: Global cluster of vulnerables exists.

Cascades are possible in “Cascade window.”

Social Contagion

Social ContagionModelsBackground

Granovetter’s model

Network version

Final size

Spreading success

Groups

References

i

ϕ = 1/3

t=4 = active at t=0

= active at t=1

= active at t=2

= active at t=3

= active at t=4

56 of 94

Threshold contagion on random networks

I Next: Find expected fractional size of spread.I Not obvious even for uniform threshold problem.I Difficulty is in figuring out if and when nodes that

need ≥ 2 hits switch on.I Problem solved for infinite seed case by Gleeson and

Cahalane:“Seed size strongly affects cascades on randomnetworks,” Phys. Rev. E, 2007. [12]

I Developed further by Gleeson in “Cascades oncorrelated and modular random networks,” Phys.Rev. E, 2008. [11]

Social Contagion

Social ContagionModelsBackground

Granovetter’s model

Network version

Final size

Spreading success

Groups

References

i

ϕ = 1/3

t=4 = active at t=0

= active at t=1

= active at t=2

= active at t=3

= active at t=4

57 of 94

Expected size of spread

Idea:I Randomly turn on a fraction φ0 of nodes at time t = 0I Capitalize on local branching network structure of

random networks (again)I Now think about what must happen for a specific

node i to become active at time t :• t = 0: i is one of the seeds (prob = φ0)• t = 1: i was not a seed but enough of i ’s friends

switched on at time t = 0 so that i ’s threshold is nowexceeded.• t = 2: enough of i ’s friends and friends-of-friends

switched on at time t = 0 so that i ’s threshold is nowexceeded.• t = n: enough nodes within n hops of i switched on

at t = 0 and their effects have propagated to reach i .

Social Contagion

Social ContagionModelsBackground

Granovetter’s model

Network version

Final size

Spreading success

Groups

References

i

ϕ = 1/3

t=4 = active at t=0

= active at t=1

= active at t=2

= active at t=3

= active at t=4

58 of 94

Expected size of spread

i

ϕ = 1/3

t=0= active,

Social Contagion

Social ContagionModelsBackground

Granovetter’s model

Network version

Final size

Spreading success

Groups

References

i

ϕ = 1/3

t=4 = active at t=0

= active at t=1

= active at t=2

= active at t=3

= active at t=4

59 of 94

Expected size of spread

i

ϕ = 1/3

t=4= active at t=0

= active at t=1

= active at t=2

= active at t=3

= active at t=4

Social Contagion

Social ContagionModelsBackground

Granovetter’s model

Network version

Final size

Spreading success

Groups

References

i

ϕ = 1/3

t=4 = active at t=0

= active at t=1

= active at t=2

= active at t=3

= active at t=4

60 of 94

Expected size of spread

Notes:I Calculations are possible if nodes do not become

inactive (strong restriction).I Not just for threshold model—works for a wide range

of contagion processes.I We can analytically determine the entire time

evolution, not just the final size.I We can in fact determine

Pr(node of degree k switching on at time t).I Asynchronous updating can be handled too.

Social Contagion

Social ContagionModelsBackground

Granovetter’s model

Network version

Final size

Spreading success

Groups

References

i

ϕ = 1/3

t=4 = active at t=0

= active at t=1

= active at t=2

= active at t=3

= active at t=4

61 of 94

Expected size of spread

Pleasantness:I Taking off from a single seed story is about

expansion away from a node.I Extent of spreading story is about contraction at a

node.

Social Contagion

Social ContagionModelsBackground

Granovetter’s model

Network version

Final size

Spreading success

Groups

References

i

ϕ = 1/3

t=4 = active at t=0

= active at t=1

= active at t=2

= active at t=3

= active at t=4

62 of 94

Expected size of spreadI Notation:φk ,t = Pr(a degree k node is active at time t).

I Notation: Bkj = Pr (a degree k node becomes activeif j neighbors are active).

I Our starting point: φk ,0 = φ0.

I(k

j

)φ j

0(1− φ0)k−j = Pr (j of a degree k node’sneighbors were seeded at time t = 0).

I Probability a degree k node was a seed at t = 0 is φ0(as above).

I Probability a degree k node was not a seed at t = 0is (1− φ0).

I Combining everything, we have:

φk ,1 = φ0 + (1− φ0)k∑

j=0

(kj

)φ j

0(1− φ0)k−jBkj .

Social Contagion

Social ContagionModelsBackground

Granovetter’s model

Network version

Final size

Spreading success

Groups

References

i

ϕ = 1/3

t=4 = active at t=0

= active at t=1

= active at t=2

= active at t=3

= active at t=4

63 of 94

Expected size of spread

I For general t , we need to know the probability anedge coming into a degree k node at time t is active.

I Notation: call this probability θt .I We already know θ0 = φ0.I Story analogous to t = 1 case. For node i :

φi,t+1 = φ0 + (1− φ0)

ki∑j=0

(ki

j

)θ j

t (1− θt )ki−jBki j .

I Average over all nodes to obtain expression for φt+1:

φt+1 = φ0 + (1− φ0)∞∑

k=0

Pk

k∑j=0

(kj

)θ j

t (1− θt )k−jBkj .

I So we need to compute θt ... massive excitement...

Social Contagion

Social ContagionModelsBackground

Granovetter’s model

Network version

Final size

Spreading success

Groups

References

i

ϕ = 1/3

t=4 = active at t=0

= active at t=1

= active at t=2

= active at t=3

= active at t=4

64 of 94

Expected size of spread

First connect θ0 to θ1:I θ1 = φ0+

(1− φ0)∞∑

k=1

kPk

〈k〉

k−1∑j=0

(k − 1

j

)θ j

0 (1− θ0)k−1−jBkj

I kPk〈k〉 = Rk = Pr (edge connects to a degree k node).

I∑k−1

j=0 piece gives Pr(degree node k activates) of itsneighbors k − 1 incoming neighbors are active.

I φ0 and (1− φ0) terms account for state of node attime t = 0.

I See this all generalizes to give θt+1 in terms of θt ...

Social Contagion

Social ContagionModelsBackground

Granovetter’s model

Network version

Final size

Spreading success

Groups

References

i

ϕ = 1/3

t=4 = active at t=0

= active at t=1

= active at t=2

= active at t=3

= active at t=4

65 of 94

Expected size of spreadTwo pieces: edges first, and then nodes

1. θt+1 = φ0︸︷︷︸exogenous

+(1− φ0)∞∑

k=1

kPk

〈k〉

k−1∑j=0

(k − 1

j

)θ j

t (1− θt )k−1−jBkj︸ ︷︷ ︸

social effects

with θ0 = φ0.2. φt+1 =

φ0︸︷︷︸exogenous

+(1− φ0)∞∑

k=0

Pk

k∑j=0

(kj

)θ j

t (1− θt )k−jBkj︸ ︷︷ ︸

social effects

.

Social Contagion

Social ContagionModelsBackground

Granovetter’s model

Network version

Final size

Spreading success

Groups

References

i

ϕ = 1/3

t=4 = active at t=0

= active at t=1

= active at t=2

= active at t=3

= active at t=4

66 of 94

Expected size of spread:

I Retrieve cascade condition for spreading from asingle seed in limit φ0 → 0.

I Depends on map θt+1 = G(θt ;φ0).I First: if self-starters are present, some activation is

assured:

G(0;φ0) =∞∑

k=1

kPk

〈k〉• Bk0 > 0.

meaning Bk0 > 0 for at least one value of k ≥ 1.I If θ = 0 is a fixed point of G (i.e., G(0;φ0) = 0) then

spreading occurs if

G′(0;φ0) =∞∑

k=0

kPk

〈k〉• (k − 1) • Bk1 > 1.

Social Contagion

Social ContagionModelsBackground

Granovetter’s model

Network version

Final size

Spreading success

Groups

References

i

ϕ = 1/3

t=4 = active at t=0

= active at t=1

= active at t=2

= active at t=3

= active at t=4

67 of 94

Expected size of spread:

In words:I If G(0;φ0) > 0, spreading must occur because some

nodes turn on for free.I If G has an unstable fixed point at θ = 0, then

cascades are also always possible.

Non-vanishing seed case:I Cascade condition is more complicated for φ0 > 0.I If G has a stable fixed point at θ = 0, and an unstable

fixed point for some 0 < θ∗ < 1, then for θ0 > θ∗,spreading takes off.

I Tricky point: G depends on φ0, so as we change φ0,we also change G.

Social Contagion

Social ContagionModelsBackground

Granovetter’s model

Network version

Final size

Spreading success

Groups

References

i

ϕ = 1/3

t=4 = active at t=0

= active at t=1

= active at t=2

= active at t=3

= active at t=4

68 of 94

General fixed point story:

00

1

1θt

θ t+1=G(θ

t;φ0)

00

1

1θt

θ t+1=G(θ

t;φ0)

00

1

1θt

θ t+1=G(θ

t;φ0)

I Given θ0(= φ0), θ∞ will be the nearest stable fixedpoint, either above or below.

I n.b., adjacent fixed points must have oppositestability types.

I Important: Actual form of G depends on φ0.I So choice of φ0 dictates both G and starting

point—can’t start anywhere for a given G.

Social Contagion

Social ContagionModelsBackground

Granovetter’s model

Network version

Final size

Spreading success

Groups

References

i

ϕ = 1/3

t=4 = active at t=0

= active at t=1

= active at t=2

= active at t=3

= active at t=4

70 of 94

Early adopters—degree distributions

t = 0 t = 1 t = 2 t = 3

0 5 10 15 200

0.05

0.1

0.15

0.2

t = 0

0 5 10 15 200

0.2

0.4

0.6

0.8

t = 1

0 5 10 15 200

0.2

0.4

0.6

0.8

t = 2

0 5 10 15 200

0.2

0.4

0.6

0.8

t = 3

t = 4 t = 6 t = 8 t = 10

0 5 10 15 200

0.1

0.2

0.3

0.4

0.5

t = 4

0 5 10 15 200

0.1

0.2

0.3

0.4

0.5

t = 6

0 5 10 15 200

0.1

0.2

0.3

0.4

t = 8

0 5 10 15 200

0.1

0.2

0.3

0.4

t = 10

t = 12 t = 14 t = 16 t = 18

0 5 10 15 200

0.05

0.1

0.15

0.2

t = 12

0 5 10 15 200

0.05

0.1

0.15

0.2

t = 14

0 5 10 15 200

0.05

0.1

0.15

0.2

t = 16

0 5 10 15 200

0.05

0.1

0.15

0.2

t = 18

Pk ,t versus k

Social Contagion

Social ContagionModelsBackground

Granovetter’s model

Network version

Final size

Spreading success

Groups

References

i

ϕ = 1/3

t=4 = active at t=0

= active at t=1

= active at t=2

= active at t=3

= active at t=4

71 of 94

The multiplier effect:

1 2 3 4 5 60

0.2

0.4

0.6

0.8

1

navg

S avg

A

1 2 3 4 5 60

1

2

3

4

navg

B

Gain

InfluenceInfluence

Averageindividuals

Top 10% individuals

Cas

cade

siz

e

Cascade size ratio

Degree ratio

I Fairly uniform levels of individual influence.I Multiplier effect is mostly below 1.

Social Contagion

Social ContagionModelsBackground

Granovetter’s model

Network version

Final size

Spreading success

Groups

References

i

ϕ = 1/3

t=4 = active at t=0

= active at t=1

= active at t=2

= active at t=3

= active at t=4

72 of 94

The multiplier effect:

1 2 3 4 5 60

0.2

0.4

0.6

0.8

1

navg

S avg

A

1 2 3 4 5 60

3

6

9

12

navg

B

Cas

cade

siz

e

InfluenceAverageIndividuals

Top 10% individuals Cascade size ratio

Degree ratio

Gain

I Skewed influence distribution example.

Social Contagion

Social ContagionModelsBackground

Granovetter’s model

Network version

Final size

Spreading success

Groups

References

i

ϕ = 1/3

t=4 = active at t=0

= active at t=1

= active at t=2

= active at t=3

= active at t=4

73 of 94

Special subnetworks can act as triggers

i0

A

B

I φ = 1/3 for all nodes

Social Contagion

Social ContagionModelsBackground

Granovetter’s model

Network version

Final size

Spreading success

Groups

References

i

ϕ = 1/3

t=4 = active at t=0

= active at t=1

= active at t=2

= active at t=3

= active at t=4

75 of 94

The power of groups...

despair.com

“A few harmless flakesworking together canunleash an avalancheof destruction.”

Social Contagion

Social ContagionModelsBackground

Granovetter’s model

Network version

Final size

Spreading success

Groups

References

i

ϕ = 1/3

t=4 = active at t=0

= active at t=1

= active at t=2

= active at t=3

= active at t=4

76 of 94

Extensions

I Assumption of sparse interactions is goodI Degree distribution is (generally) key to a network’s

functionI Still, random networks don’t represent all networksI Major element missing: group structure

Social Contagion

Social ContagionModelsBackground

Granovetter’s model

Network version

Final size

Spreading success

Groups

References

i

ϕ = 1/3

t=4 = active at t=0

= active at t=1

= active at t=2

= active at t=3

= active at t=4

77 of 94

Group structure—Ramified random networks

p = intergroup connection probabilityq = intragroup connection probability.

Social Contagion

Social ContagionModelsBackground

Granovetter’s model

Network version

Final size

Spreading success

Groups

References

i

ϕ = 1/3

t=4 = active at t=0

= active at t=1

= active at t=2

= active at t=3

= active at t=4

78 of 94

Bipartite networks

c d ea b

2 3 41

a

b

c

d

e

contexts

individuals

unipartitenetwork

Social Contagion

Social ContagionModelsBackground

Granovetter’s model

Network version

Final size

Spreading success

Groups

References

i

ϕ = 1/3

t=4 = active at t=0

= active at t=1

= active at t=2

= active at t=3

= active at t=4

79 of 94

Context distance

eca

high schoolteacher

occupation

health careeducation

nurse doctorteacherkindergarten

db

Social Contagion

Social ContagionModelsBackground

Granovetter’s model

Network version

Final size

Spreading success

Groups

References

i

ϕ = 1/3

t=4 = active at t=0

= active at t=1

= active at t=2

= active at t=3

= active at t=4

80 of 94

Generalized affiliation model

100

eca b d

geography occupation age

0

(Blau & Schwartz, Simmel, Breiger)

Social Contagion

Social ContagionModelsBackground

Granovetter’s model

Network version

Final size

Spreading success

Groups

References

i

ϕ = 1/3

t=4 = active at t=0

= active at t=1

= active at t=2

= active at t=3

= active at t=4

81 of 94

Generalized affiliation model networks withtriadic closure

I Connect nodes with probability ∝ exp−αd

whereα = homophily parameterandd = distance between nodes (height of lowestcommon ancestor)

I τ1 = intergroup probability of friend-of-friendconnection

I τ2 = intragroup probability of friend-of-friendconnection

Social Contagion

Social ContagionModelsBackground

Granovetter’s model

Network version

Final size

Spreading success

Groups

References

i

ϕ = 1/3

t=4 = active at t=0

= active at t=1

= active at t=2

= active at t=3

= active at t=4

82 of 94

Cascade windows for group-based networks

Gen

eral

ized

Affili

atio

n

A

Gro

up n

etwo

rks

Single seed Coherent group seed

Mod

el n

etwo

rks

Random set seed

Rand

om

F

C

D E

B

Social Contagion

Social ContagionModelsBackground

Granovetter’s model

Network version

Final size

Spreading success

Groups

References

i

ϕ = 1/3

t=4 = active at t=0

= active at t=1

= active at t=2

= active at t=3

= active at t=4

83 of 94

Multiplier effect for group-based networks:

4 8 12 16 200

0.2

0.4

0.6

0.8

1

navg

S avg

A

4 8 12 16 200

1

2

3

navg

B

0 4 8 12 160

0.2

0.4

0.6

0.8

1

navg

S avg

C

0 4 8 12 160

1

2

3

navg

D

Degree ratio

Gain

Cascadesize ratio

Cascadesize ratio < 1!

I Multiplier almost always below 1.

Social Contagion

Social ContagionModelsBackground

Granovetter’s model

Network version

Final size

Spreading success

Groups

References

i

ϕ = 1/3

t=4 = active at t=0

= active at t=1

= active at t=2

= active at t=3

= active at t=4

84 of 94

Assortativity in group-based networks

0 5 10 15 200

0.2

0.4

0.6

0.8

k

0 4 8 120

0.5

1

k

AverageCascade size

Local influence

Degree distributionfor initially infected node

I The most connected nodes aren’t always the most‘influential.’

I Degree assortativity is the reason.

Social Contagion

Social ContagionModelsBackground

Granovetter’s model

Network version

Final size

Spreading success

Groups

References

i

ϕ = 1/3

t=4 = active at t=0

= active at t=1

= active at t=2

= active at t=3

= active at t=4

85 of 94

Social contagion

SummaryI ‘Influential vulnerables’ are key to spread.I Early adopters are mostly vulnerables.I Vulnerable nodes important but not necessary.I Groups may greatly facilitate spread.I Seems that cascade condition is a global one.I Most extreme/unexpected cascades occur in highly

connected networksI ‘Influentials’ are posterior constructs.I Many potential influentials exist.

Social Contagion

Social ContagionModelsBackground

Granovetter’s model

Network version

Final size

Spreading success

Groups

References

i

ϕ = 1/3

t=4 = active at t=0

= active at t=1

= active at t=2

= active at t=3

= active at t=4

86 of 94

Social contagion

ImplicationsI Focus on the influential vulnerables.I Create entities that can be transmitted successfully

through many individuals rather than broadcast fromone ‘influential.’

I Only simple ideas can spread by word-of-mouth.(Idea of opinion leaders spreads well...)

I Want enough individuals who will adopt and display.I Displaying can be passive = free (yo-yo’s, fashion),

or active = harder to achieve (political messages).I Entities can be novel or designed to combine with

others, e.g. block another one.

Social Contagion

Social ContagionModelsBackground

Granovetter’s model

Network version

Final size

Spreading success

Groups

References

i

ϕ = 1/3

t=4 = active at t=0

= active at t=1

= active at t=2

= active at t=3

= active at t=4

87 of 94

References I

[1] A. Bentley, M. Earls, and M. J. O’Brien.I’ll Have What She’s Having: Mapping SocialBehavior.MIT Press, Cambridge, MA, 2011.

[2] S. Bikhchandani, D. Hirshleifer, and I. Welch.A theory of fads, fashion, custom, and culturalchange as informational cascades.J. Polit. Econ., 100:992–1026, 1992.

[3] S. Bikhchandani, D. Hirshleifer, and I. Welch.Learning from the behavior of others: Conformity,fads, and informational cascades.J. Econ. Perspect., 12(3):151–170, 1998. pdf ()

Social Contagion

Social ContagionModelsBackground

Granovetter’s model

Network version

Final size

Spreading success

Groups

References

i

ϕ = 1/3

t=4 = active at t=0

= active at t=1

= active at t=2

= active at t=3

= active at t=4

88 of 94

References II

[4] J. M. Carlson and J. Doyle.Highly optimized tolerance: A mechanism for powerlaws in designed systems.Phys. Rev. E, 60(2):1412–1427, 1999. pdf ()

[5] J. M. Carlson and J. Doyle.Highly optimized tolerance: Robustness and designin complex systems.Phys. Rev. Lett., 84(11):2529–2532, 2000. pdf ()

[6] N. A. Christakis and J. H. Fowler.The spread of obesity in a large social network over32 years.New England Journal of Medicine, 357:370–379,2007. pdf ()

Social Contagion

Social ContagionModelsBackground

Granovetter’s model

Network version

Final size

Spreading success

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i

ϕ = 1/3

t=4 = active at t=0

= active at t=1

= active at t=2

= active at t=3

= active at t=4

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References III

[7] N. A. Christakis and J. H. Fowler.The collective dynamics of smoking in a large socialnetwork.New England Journal of Medicine, 358:2249–2258,2008. pdf ()

[8] R. B. Cialdini.Influence: Science and Practice.Allyn and Bacon, Boston, MA, 4th edition, 2000.

[9] J. H. Fowler and N. A. Christakis.Dynamic spread of happiness in a large socialnetwork: longitudinal analysis over 20 years in theFramingham Heart Study.BMJ, 337:article #2338, 2008. pdf ()

Social Contagion

Social ContagionModelsBackground

Granovetter’s model

Network version

Final size

Spreading success

Groups

References

i

ϕ = 1/3

t=4 = active at t=0

= active at t=1

= active at t=2

= active at t=3

= active at t=4

90 of 94

References IV

[10] M. Gladwell.The Tipping Point.Little, Brown and Company, New York, 2000.

[11] J. P. Gleeson.Cascades on correlated and modular randomnetworks.Phys. Rev. E, 77:046117, 2008. pdf ()

[12] J. P. Gleeson and D. J. Cahalane.Seed size strongly affects cascades on randomnetworks.Phys. Rev. E, 75:056103, 2007. pdf ()

[13] M. Granovetter.Threshold models of collective behavior.Am. J. Sociol., 83(6):1420–1443, 1978. pdf ()

Social Contagion

Social ContagionModelsBackground

Granovetter’s model

Network version

Final size

Spreading success

Groups

References

i

ϕ = 1/3

t=4 = active at t=0

= active at t=1

= active at t=2

= active at t=3

= active at t=4

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References V

[14] M. Granovetter and R. Soong.Threshold models of diversity: Chinese restaurants,residential segregation, and the spiral of silence.Sociological Methodology, 18:69–104, 1988. pdf ()

[15] M. S. Granovetter and R. Soong.Threshold models of interpersonal effects inconsumer demand.J. Econ. Behav. Organ., 7:83–99, 1986. pdf ()

[16] E. Katz and P. F. Lazarsfeld.Personal Influence.The Free Press, New York, 1955.

[17] T. Kuran.Now out of never: The element of surprise in theeast european revolution of 1989.World Politics, 44:7–48, 1991. pdf ()

Social Contagion

Social ContagionModelsBackground

Granovetter’s model

Network version

Final size

Spreading success

Groups

References

i

ϕ = 1/3

t=4 = active at t=0

= active at t=1

= active at t=2

= active at t=3

= active at t=4

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References VI

[18] T. Kuran.Private Truths, Public Lies: The SocialConsequences of Preference Falsification.Harvard University Press, Cambridge, MA, Reprintedition, 1997.

[19] T. C. Schelling.Dynamic models of segregation.J. Math. Sociol., 1:143–186, 1971. pdf ()

[20] T. C. Schelling.Hockey helmets, concealed weapons, and daylightsaving: A study of binary choices with externalities.J. Conflict Resolut., 17:381–428, 1973. pdf ()

[21] T. C. Schelling.Micromotives and Macrobehavior.Norton, New York, 1978.

Social Contagion

Social ContagionModelsBackground

Granovetter’s model

Network version

Final size

Spreading success

Groups

References

i

ϕ = 1/3

t=4 = active at t=0

= active at t=1

= active at t=2

= active at t=3

= active at t=4

93 of 94

References VII

[22] D. Sornette.Critical Phenomena in Natural Sciences.Springer-Verlag, Berlin, 1st edition, 2003.

[23] D. J. Watts.A simple model of global cascades on randomnetworks.Proc. Natl. Acad. Sci., 99(9):5766–5771, 2002.pdf ()

[24] D. J. Watts and P. S. Dodds.Influentials, networks, and public opinion formation.Journal of Consumer Research, 34:441–458, 2007.pdf ()

Social Contagion

Social ContagionModelsBackground

Granovetter’s model

Network version

Final size

Spreading success

Groups

References

i

ϕ = 1/3

t=4 = active at t=0

= active at t=1

= active at t=2

= active at t=3

= active at t=4

94 of 94

References VIII

[25] D. J. Watts and P. S. Dodds.Threshold models of social influence.In P. Hedström and P. Bearman, editors, The OxfordHandbook of Analytical Sociology, chapter 20,pages 475–497. Oxford University Press, Oxford,UK, 2009. pdf ()

[26] U. Wilensky.Netlogo segregation model.http://ccl.northwestern.edu/netlogo/models/Segregation. Center for ConnectedLearning and Computer-Based Modeling,Northwestern University, Evanston, IL., 1998.